Approximation related to quotient functionals

作者：Setzer S; Steidl G^*; Teuber T; Moerkotte G

来源：Journal of Approximation Theory, 2010, 162(3): 545-558.

DOI：10.1016/j.jat.2009.08.009

• Best approximation
• Polynomial approximation
• Chebyshev sets
• Convex optimization
• Second order cone programming

We examine the best approximation of componentwise positive vectors or positive continuous functions f by linear combinations (f) over cap = Sigma(j)alpha(j)phi(j) of given vectors or functions phi(j) with respect to atnctionals Q(p), 1 <= p <= infinity, involving quotients max {f/(f) over cap.(f) over cap /f) rather than differences vertical bar f - (f) over cap vertical bar. We verify the existence of a best approximating function under mild conditions on {phi(j)}(j=1)(n). For discrete data, we compute a best approximating function with respect to Q(p), p = 1, 2, infinity by second order cone programming. Special attention is paid to the Q(infinity), functional in both the discrete and the continuous setting. Based on the computation of the subdifferential of our convex functional Q(infinity) we give an equivalent characterization of the hest approximation by using its extremal set. Then we apply this characterization to prove the uniqueness of the best Q(infinity) approximation for Chebyshev sets {phi(j)}(j=1)(n).

• 出版日期2010-3